Wash and scrub the beets, and trim off the greens. Peel the beets and, using a hand guard, carefully slice them paper-thin on a mandoline. Cover the sliced beets with a wet paper towel while you prepare the rest of the salad.
In a medium bowl, whisk together the white balsamic vinegar and 2 tablespoons of the olive oil and season with salt and pepper. Add the arugula and toss until evenly coated.
Set out 4 small (salad-sized) plates. Create a circle of beets on each plate, alternating colors between golden and striped (or red). Drizzle the remaining 1 tablespoon of olive oil on the beets and season with sea salt and pepper. Place a pile of arugula in the center of each circle and top with goat cheese and hazelnuts.